The domain decomposition methods considered are preconditioned conjugate gradient methods designed for the very large algebraic systems of equations which often arise in finite element practice. They are designed for massively parallel computer systems and the preconditioners are built from solvers on the substructures into whichthe domain of the given problem is partitioned. In addition, to obtain scalability, there must be a coarse problem, with a small number of degrees of freedom for each substructure. The design of this coarse problem is crucial for obtaining rapidly convergent iterations and poses the most interesting challenge in the analysis.Our work will be illustrated by overlapping Schwarz methods for almost incompressible elasticity approximated by mixed finite element and mixed spectral element methods. These algorithms is now used extensively at the SANDIA, Albuquerque laboratories and were developed in close collaboration with Dr. Clark R. Dohrmann. These results illustrate two roles of the coarse component of the preconditioner.Currently, these algorithms are being actively developed for problems posed in H(curl) and H(div). This work requires the development of new coarse spaces. We will also comment on recent work on extending domain decomposition theory to subdomains with quite irregular boundaries. This work is relevant because of the use of mesh partitioners in the decomposition of large finite element matrices.